CN112882391B - Double-end event triggered nonlinear control method - Google Patents
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Abstract
The invention discloses a double-end event-triggered nonlinear control method, wherein two self-adaptive event generators are respectively installed in network channels from a sensor to an observer and from the observer to a controller to ensure the smooth communication of a global network, an event trigger threshold value is obtained by utilizing a self-adaptive function related to the current state error, and a special time interval division method is adopted to unify the time-varying delay information of the network channels at the two sides of the observer under a frame; the method comprises the steps of providing an observer-based non-parallel distributed compensation fuzzy feedback controller to ensure the stability of a networked system, supplementing related information of double-end adaptive event triggering conditions in a Lyapunov function, obtaining the stability condition of the system by applying a Lyapunov theoretical analysis method, and developing a linear matrix inequality-based membership function related stability condition under an incomplete precondition matching condition to further obtain a parameter matrix of an observer and a controller in the system.
Description
Technical Field
The invention belongs to the technical field of event trigger mechanisms in network control systems, and particularly relates to a double-end event-triggered nonlinear control method.
Background
In recent years, due to the popularization of the internet, a network control system that is a combination of a conventional control method and a network has gradually become one of the major research hotspots in the control field. Different from the traditional control system structure, the network control system utilizes a network transmission channel to replace a complex wired connection to transmit control information and state information so as to simplify the physical complexity of communication and realize long-distance transmission, and has higher control efficiency and lower maintenance cost. However, when data is transmitted in a large scale, network congestion may occur and control performance may be affected due to the limitation of network equipment load. To solve this problem, an event triggering scheme has been proposed to improve the network transmission environment, which is periodically adopted and is characterized in that a necessary sampling signal is transmitted to reduce the network load only if an event triggering condition is satisfied, which makes it more desirable than periodic sampling. Generally, the structure and state of a system may be different with time, a conventional event triggering mechanism depends on a preset constant threshold, and an event triggering condition based on a stable threshold is obviously not an optimal choice, which cannot effectively reduce network congestion and ensure optimal system performance.
The adaptive event triggering mechanism can be adaptively adjusted according to the current network condition so as to effectively ensure the balance between the network load and the system performance, the control problem of a pure feedback object is solved by using the adaptive event triggering mechanism, and a wider system partial derivative condition is obtained. When network packet loss and actuator faults are considered, the waste of communication resources is reduced by providing an adaptive event triggering mechanism. In order to save computational resources effectively, a novel adaptive event triggering mechanism is proposed, which introduces an adaptive function related to the state error to adjust the threshold. Finally, the reliability of the proposed theory is proved by comparing with the simulated data transmission efficiency.
Due to the complex and nonlinear nature of the structure, networked control systems are often difficult to model and perform accurate mathematical analyses. The Takagi-Sugeno (T-S) model consists of a linear subsystem, provides a mathematical basis and theoretical guarantees for solving the problem, and is beneficial to deriving the stability condition and the controller of the nonlinear system. In general, due to the complex structure of the system, some states cannot be obtained by measurement. Therefore, observer-based controller design has become an active research focus for researchers. The conclusions provided by the observer-based controller above are based on a single-channel event-triggered approach, however, given the more complex engineering issues, there are situations where both the observer input and output signals need to be communicated over a network, so a good transmission environment must be created for both sensor-to-observer and observer-to-controller channels of data through a suitable event-triggering mechanism.
In recent years, the event trigger mechanism has become one of the important means for dealing with communication data redundancy in the networked control system, and is mainly developing towards two directions: one is to apply the event trigger mechanism to a more complex network environment, and the other is to improve the existing event trigger mechanism, so that the system can improve the data transmission efficiency when meeting the expected performance. The method considers that network channels are constructed at both ends of the observer, and under the framework of incomplete premise matching, the problems of observer-based fuzzy controller design and stability of a networked control system with a double-end adaptive event triggering mechanism are considered.
Disclosure of Invention
In order to overcome the defects in the prior art, the nonlinear control method triggered by the double-end event provided by the invention considers that network channels are constructed at both ends of the observer, and considers the problems of observer-based fuzzy controller design and stability of a networked control system with a double-end adaptive event triggering mechanism under the framework of incomplete premise matching.
In order to achieve the purpose of the invention, the invention adopts the technical scheme that: a double-ended event triggered nonlinear control method comprises the following steps:
s1, constructing a nonlinear networked control system closed-loop model with a double-end event trigger mechanism;
s2, defining a Lyapunov function containing event triggering information, and determining the stability condition of a closed-loop model of the nonlinear networked control system by using the Lyapunov function;
and S3, solving parameter matrixes of an observer and a controller of the closed-loop model of the nonlinear networked control system through a linear matrix inequality based on the stability condition of the closed-loop model of the nonlinear networked control system, and further realizing nonlinear control.
The invention has the beneficial effects that:
1) the current discussion of event triggering mechanisms remains on single channel solutions. However, in order to meet the increasing engineering requirements, control signals sometimes need to be transmitted to the actuators through the network, so that it is of great significance to discuss a double-end event triggering mechanism for guaranteeing network smoothness of a global system. The invention provides a novel double-end self-adaptive event triggering scheme, wherein two self-adaptive event generators are respectively installed in a sensor-observer channel and an observer-controller channel to ensure smooth communication of a global network, and an event triggering threshold value is obtained by utilizing a self-adaptive function related to a current state error.
2) Because the introduction of the dual-end network can cause different time delays to be generated when communication data are transmitted in different network channels, time-varying time delay information of the network channels on two sides of the observer needs to be considered respectively, and for convenience of analysis, a special time interval division method is adopted to unify the time delay information under a framework.
3) The invention provides an observer-based non-parallel distributed compensation fuzzy feedback controller to ensure the stability of a networked system, relevant information of double-end adaptive event triggering conditions is supplemented in a Lyapunov function, the stability condition of the system is obtained by applying a Lyapunov theoretical analysis method, the stability condition related to a membership function based on a linear matrix inequality is developed under the incomplete precondition matching condition, a relaxation matrix with proper dimensionality is designed according to partial information of the membership function, and the conservatism of the proposed theoretical result is further reduced.
Drawings
Fig. 1 is a flowchart of a double-ended event triggered nonlinear control method provided by the present invention.
Fig. 2 is a structural block diagram of a closed-loop model of a nonlinear networked control system provided by the present invention.
Fig. 3 is a schematic structural diagram of a permanent magnet synchronous motor model in an embodiment of the present invention.
Fig. 4 is a schematic diagram of a system state x (t) in an embodiment of the present invention.
Fig. 6 is a schematic diagram of network transmission in a sensor-to-controller channel under an adaptive event triggering mechanism according to an embodiment of the present invention.
Fig. 7 is a schematic diagram of network transmission in an observer to controller channel under an adaptive event triggering mechanism according to an embodiment of the present invention.
Fig. 8 is a schematic diagram of network transmission in a sensor-to-controller channel under a conventional event trigger mechanism according to an embodiment of the present invention.
Fig. 9 is a schematic diagram of network transmission in an observer-to-controller channel under a conventional event trigger mechanism according to an embodiment of the present invention.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate the understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and it will be apparent to those skilled in the art that various changes may be made without departing from the spirit and scope of the invention as defined and defined by the appended claims, and all changes that can be made by the invention using the inventive concept are intended to be protected.
Example 1:
as shown in fig. 1, a double-ended event triggered nonlinear control method includes the following steps:
s1, constructing a nonlinear networked control system closed-loop model with a double-end event trigger mechanism;
s2, defining a Lyapunov function containing event triggering information, and determining the stability condition of a closed-loop model of the nonlinear networked control system by using the Lyapunov function;
and S3, solving parameter matrixes of an observer and a controller of the closed-loop model of the nonlinear networked control system through a linear matrix inequality based on the stability condition of the closed-loop model of the nonlinear networked control system, and further realizing nonlinear control.
As shown in fig. 2, the nonlinear networked control system closed-loop model in step S1 includes a controlled object, a sensor, an observer, a controller, and an actuator, which are sequentially connected in a closed-loop manner;
the output end of the controller is also connected with the input end of the observer;
to facilitate stability analysis, the controlled object in FIG. 2 is described by a T-S fuzzy model;
in order to reduce unnecessary data transmission, the observer is an observer with a double-end self-adaptive event trigger mechanism;
the controller is a fuzzy controller based on a non-parallel distributed compensation strategy.
In order to ensure the stability of the system, the controller is a fuzzy controller based on non-parallel distribution compensation measurement.
In this embodiment, for the controlled object, the T-S fuzzy model with r fuzzy rules is described as follows:
for the object rule i, when the controlled object blurs the precondition f1(x(t))…fκ(x (t)) the corresponding fuzzy sets are in turnThen, obtaining:
in the formula, r is the number of fuzzy rules;for fuzzy precondition variable fq(x (t)) a corresponding fuzzy set, where q 1, 2.. and k and i 1, 2.. and r, x (t) denote state vectors, u (t) denotes control input vectors, a (a) denotes control input vectors, and (b) denotes control input vectorsi,BiAnd CiAre all a matrix of the system, and,is a state rate of change vector, y (t) is a system output vector, for the purposes of an n-dimensional euclidean space, is an m-dimensional euclidean space;
the global model corresponding to the controlled object is as follows:
in the formula, wi(x (t)) is degree of membershipCorresponding fuzzy weight in the controlled object;
membership function wi(x (t)) satisfies:
the observer with the double-end adaptive event triggering mechanism in this embodiment ensures smooth operation of network communication and improves network transmission efficiency, and in order to simplify analysis, the observer in this embodiment is described in the following manner:
for observer rule j, when observer blurs the preconditionsThe corresponding fuzzy sets are sequentiallyThen, obtaining:
in the formula (I), the compound is shown in the specification,in order to observe the rate of change of state vector,is a state vector of an observer, anu (t) is a control input vector, observer rule j satisfies j is more than or equal to 2 and less than or equal to r,is a fuzzy precondition of the observerA corresponding fuzzy set, q 1,2jAs observer parameters, Aj,BjAnd CjAre all a matrix of the system, and,is the output vector through the first event generator;
the global model corresponding to the observer is as follows:
in the formula (I), the compound is shown in the specification,is degree of membershipA corresponding fuzzy weight in the observer;
in this embodiment, in order to make the structural design of the controller more flexible, we consider an independent precondition variable in the fuzzy rule so as to optimize the structural design of the control, the controller is a fuzzy controller based on a non-parallel distribution compensation strategy, and the description mode of the controller is as follows:
for controller rule l, when the controller fuzzes the precondition variablesThe corresponding fuzzy sets are sequentiallyThen, obtaining:
wherein u (t) is a control input vector,fuzzy precondition variables for a controllerCorresponding fuzzy sets, q 1,2, p, l 1,2, r, p is the number of fuzzy controller preconditions, Kl is a fuzzy controller parameter matrix,inputting a state vector for a controller;
the global model corresponding to the controller is as follows:
in the formula (I), the compound is shown in the specification,is the fuzzy weight of the fuzzy controller;
the error of the observer is:
based on the controlled object, the observer and the global models (2), (5) and (10) corresponding to the controller, the closed-loop model of the nonlinear network control system is obtained as follows:
therein, Ψijl、Aijl、Lijl、LeijlAnd LeyjIntermediate parameters in a closed-loop model of the nonlinear network control system are controlled.
In the embodiment, in a closed-loop model of a nonlinear networked control system, under an adaptive event triggering mechanism, a first event generator is arranged in a data transmission channel from a sensor to an observer, and the first event generator can reduce network redundancy signals;
in the sensor-to-observer data path, at event trigger timesSystem output signal transmitted by triggering first event generator through sensor end eventAt the moment of timeThe data center of the observer is reached through a transmission network; wherein the content of the first and second substances,is a time of dayThe network at the sensor end of (1) induces a time delay,τyis a time of dayThe maximum network induced delay of the sensor end;
the adaptive event trigger mechanism determines the current signal based on adaptive event trigger conditionsWhether or not to be transmitted to the observer, the next event triggering instant when the adaptive event triggering condition is satisfiedComprises the following steps:
in the formula (I), the compound is shown in the specification,j is the number of consecutive packets that occur for jh sets that satisfy the condition, for the maximum allowable amount of data packet loss to continuously occur,is the difference between the current output signal and the last trigger time, ΩyIs an event trigger matrix, and Ωy>0,ey(T) is the error, χ, between the current output signal of the T-S fuzzy model and the last transmitted output signaly(t) is satisfying xy(t)∈(0,1]Variable trigger threshold of yT() means for outputting a vector for the current system, y (-) is the current system output vector;
wherein the error e of the current output signal of the T-S fuzzy model and the last transmitted output signaly(t) is:
variable trigger threshold χy(t) satisfies:
wherein the content of the first and second substances,in order for the event to trigger the threshold value,for adjusting the variable trigger threshold χy(t) a parameter of convergence speed; therefore, the event trigger condition can be dynamically adjusted through the output signal of the system;
considering network induced delay, based on the characteristics of an adaptive event trigger mechanism, the input signal of an observerComprises the following steps:
based on the adaptive event triggering mechanism condition, the input signal of the observer in two adjacent triggering instant intervals is a constant, which effectively saves the occupation of communication resources, and for the convenience of analysis, the time interval with network-induced delayIs divided into:
wherein, the first and the second end of the pipe are connected with each other,in order to be able to process the time interval,
defining a time-varying function d (t):
therefore, the observer input signal based on the adaptive event triggering mechanism is:
wherein d (t) is time-varyingThe function of the function is that of the function, in order to be able to process the time interval,epsilon is an intermediate parameter which is,tau is the maximum network-induced time delay,
in the embodiment, a second event generator is arranged in a data transmission channel from the observer to the controller;
in the data transmission channel from the observer to the controller, at the moment of application of the triggerObserver state signal transmitted by observer-end data trigger transmitterAt the time of dayThe data center of the controller is reached through a transmission network;is a time of dayThe maximum network at the observer end of (1) induces a time delay, is a time of dayThe maximum network induced delay of the observer end;
the second event generator reduces the network redundancy of the state signal under the condition of an adaptive event triggering mechanism according to the state information of the observer, wherein the condition of the adaptive triggering mechanism is as follows:
in the formula (I), the compound is shown in the specification,for the moment when the next event trigger occurs,as a transpose of the difference of the current observer state and the last triggered observer state,in order for the event to trigger the matrix,the difference between the current observer state and the last triggered observer state,to adapt the trigger threshold of the trigger mechanism,is a transpose of the state vector of the current observer,is the current observer state vector;
in the formula (I), the compound is shown in the specification,in order for the threshold to be triggered by an event,for adjusting trigger thresholdA parameter of convergence speed; input signal of controller based on adaptive event trigger mechanismComprises the following steps:
to facilitate the analysis of asynchronous information of the sensor-to-observer and observer-to-controller channels, the subintervals of the two network channel transmission times need to be unified within one framework,indicates at the current sampling instant isLast event triggering instant of time observer to transmitter in controller channelDifference between present status signal and last transmitted status signalComprises the following steps:
the same method as the division of the time interval from the sensor to the observer channel, the time intervalIt can also be divided into time sets in units of time length h, resulting in:
thus, the time interval between two adjacent transmitters in the observer to controller can be expressed as:
thus, equation (22) can be converted to:
in summary, the input signals of the controllers based on the adaptive event triggering mechanisms according to (16), (19) and (23)Comprises the following steps:
it should be noted that the maximum network-induced delay of the dual-end communication network in this embodiment is defined asThe data center at the sensor end is supposed to detect whether the buffer of the data center needs to be updated at each moment ih + tau, (i e N), and the data center at the observer end is supposed to detect whether the buffer of the data center needs to be updated at each moment ih + tau, (i e N); furthermore, we ignore the computation time required to update the data. By the method, asynchronous network time delays of a sensor-observer channel and an observer-controller channel can be unified in a framework so as to analyze the stability of the system.
Step S2 of this embodiment specifically includes:
s21, defining a Lyapunov function containing event trigger information;
s22, carrying out derivation on the Lyapunov function, and eliminating an integral term in the Lyapunov function to obtain the stability condition of the closed-loop model of the nonlinear networked control system;
s23, eliminating a nonlinear term in the stability condition of the nonlinear networked control system closed-loop model;
and S24, on the basis of the stability condition of the networked control system closed-loop model with the nonlinear term eliminated, obtaining a system stability condition related to the membership function based on the linear matrix inequality by using the time delay information of the membership function, and taking the system stability condition as the final stability condition of the nonlinear networked control system closed-loop model.
In the above step S22, dM>0,Andif there is a matrix P with suitable dimensions>0,Q>0,R>0,W>0,Ωy>0,And U, the stability condition of the closed-loop model of the nonlinear networked control system is as follows:
in the formula phiijlIn order to determine the matrix for the stability,andare all phiijlIs an element ofA transposed matrix of mid-diagonal positions;
wherein:
F=[Aijl L ijl 0 Leijl Leyj]
in the formula (I), the compound is shown in the specification,andare all made ofElement of (1), col { ·The matrix is a column matrix, the diag { · } is a block diagonal matrix, and F is an auxiliary matrix;
and is
In the formula, G ═ I I, E ═ I0, and I is a unit matrix with an appropriate dimension.
The proof process that the system can be kept stable using the above-described stability condition based on the lyapunov function in step S21 is:
the Lyapunov function is chosen to be:
V(t)=V1(t)+V2(t)+V3(t) (29)
wherein ξT(t) is the transpose of the coupling vector for observed state change rate and state error, ξ (t) is the coupling vector for observed state change rate and state error, P, Q, R and W are both matrices with appropriate dimensions, ikh is an intermediate parameter which is a function of,dMis the upper delay bound.
Obtained by derivation of the aforementioned lyapunov function (29):
considering a network communication protocol (12) of the sensor-to-observer channel and an adaptive function (14) of the trigger threshold, we obtain:
taking into account the network communication protocol (19) of the observer to controller channel and the adaptive function (20) of the trigger threshold, we obtain:
based on theorem 1, obtaining
Wherein:
by combining (29) to (31), we can obtain:
according to the Shu's complement theory, the following results are obtained:
therefore, if there is Φijl<0, then the system (11) is asymptotically stable.
Because the existence of the nonlinear term, the stability condition cannot obtain a specific observer parameter matrix and a specific controller parameter matrix, so to solve these parameters, we obtain the system stability condition related to the membership function based on the linear matrix inequality in step S24 by using a linear matrix inequality method, where the stability condition is:
when d isM>0,Andand the membership function of the fuzzy controller satisfiesWherein 0<ιl≤1,ιlFor appropriate constants, if there is a matrix X with suitable dimensions>0, TjAnd YlThen, the system stability condition related to the membership function based on the linear matrix inequality is:
wherein
The stability condition can keep the system stable, and the proving process for determining the parameter matrix of the controller and the parameter matrix of the observer is specifically as follows:
Definition of
Multiplying the matrix T on the left and right sides of (28) to obtain
Wherein
Due to the fact that
To obtain
Thus, we can get:
Thus, the following results:
It should be noted that we useSimplifyAnd, the matrix X is,and T are both well-designed to linearize the nonlinear term in theory 2. In order to reduce the conservatism of the derived result, the stability condition of the linear matrix inequality is obtained by considering the boundary conditions of the membership functions of the fuzzy observer and the fuzzy controller.
theorem 1: considering satisfaction of conditionsτ (t) of (1). For any satisfied conditionOf (2) matrixAnd U ∈ Rn×nThe following equations are all true:
theorem 2: for a full rank matrix rank (c) m,the singular value decomposition of C can be described as C ═ O [ S0 ═ O]VTWherein O.OTIs equal to I and V.VTI. Let matrix X>0,Andif presentSatisfy the requirements ofThen the following holds:
wherein X >0 represents a true symmetric positive definite matrix. I denotes an identity matrix of a suitable dimension
The parameter matrix of the observer in step S3 is obtained based on the stability condition as:
the parameter matrix of the controller is:
Kl=YlX-1
example 2:
this embodiment provides a simulation example based on the method in embodiment 1 above:
as shown in fig. 3, the permanent magnet synchronous motor converts the feedback current signal by using a rotor reference system, and a mathematical model of the permanent magnet synchronous motor is expressed as follows:
in the formula, vmAnd vnRepresenting the voltage in the reference frame of the rotor, imAnd inIn the rotor reference systemCurrent, omegasThe motor speed is shown, and the rest physical parameters refer to the table 1.
To simplify the design of the control rate, assume that the m-axis current is 0. Then, the electromagnetic torque is adjusted using the n-axis current, which can be expressed as follows:
the dynamic equations can be expressed as follows:
wherein T isLRepresenting the input torque, the remaining physical parameters refer to table 1.
Table 1: model parameters of permanent magnet synchronous motor
In order to simplify the problem, the influence of external interference on the permanent magnet synchronous motor is ignored. And order statesThe permanent magnet synchronous motor system can be modeled as a two-rule T-S fuzzy system:
system rule 1: if x1(t) is M1, then
System rule 2: if x1(t) is M2Then, then
Wherein:
the membership function is:
wherein the content of the first and second substances,represents omegasIs determined by the minimum estimate of (c) of,represents omega s2, d2 is 10 and ω is the maximum estimate of (d 1) ═ 2, d2 ═ 10s∈[d1,d2]。
Order toAnd dM 0.07, the following parameter matrix can be obtained according to theory 2 (resulting in system stability conditions related to membership functions based on linear matrix inequalities):
Ωy=8.9353
L1=[-3.6755 0.0989 0.0171]T
L2=[-1.7078 0.0124 0.0480]T
let the sampling period h equal to 0.02, initial state x0=[10.5 -9.1 0.7]TAnd an initial threshold valueBy MATLAB simulation, the permanent magnet synchronous motor can be effectively controlled, and network communication resources are saved. FIG. 4 shows that the state of the system can converge quickly to the origin; FIG. 5 illustrates state errors of the observer and system; FIGS. 6 and 7 show event triggering scenarios in the sensor-observer and observer-controller channels, respectively; FIGS. 8 and 9 illustrate the use of a conventional event trigger mechanism and event trigger threshold σ, respectivelyy=0.2,Event triggering conditions in a sensor-to-observer channel and an observer-to-controller channel; tables 2 and 3 list the number of network communication data transmissions under different communication schemes. Wherein table 2 lists the amount of data transferred from the sensor to the observer channel and table 3 lists the amount of data transferred from the observer to the controller channel.
Table 2: data volume transmitted under different communication schemes in sensor-observer channel
Table 3: data volume transmission under different communication schemes in observer-controller channel
Claims (7)
1. A double-end event triggered nonlinear control method is characterized by comprising the following steps:
s1, constructing a nonlinear networked control system closed-loop model with a double-end event trigger mechanism;
s2, defining a Lyapunov function containing event triggering information, and determining the stability condition of a closed-loop model of the nonlinear networked control system by using the Lyapunov function;
s3, solving parameter matrixes of an observer and a controller of the nonlinear networked control system closed-loop model through a linear matrix inequality based on the stability condition of the nonlinear networked control system closed-loop model, and further realizing nonlinear control;
the nonlinear networked control system closed-loop model in the step S1 includes a controlled object, a sensor, an observer, a controller and an actuator, which are sequentially connected in a closed-loop manner;
the output end of the controller is also connected with the input end of the observer;
the controlled object is described by a T-S fuzzy model;
the observer is an observer with a double-end self-adaptive event trigger mechanism;
the controller is a fuzzy controller based on a non-parallel distribution compensation strategy;
the T-S fuzzy model with r fuzzy rules is described in the following manner:
for the object rule i, when the controlled object blurs the precondition variable f1(x(t))…fκ(x (t)) the corresponding fuzzy sets are sequentiallyThen, obtaining:
in the formula, r is the number of fuzzy rules;for fuzzy precondition variable fq(x (t)) corresponding fuzzy sets, where q ═ 1, 2., κ and i ═ 1, 2., r, x (t) denote state vectors, u (t) denote control input vectors, a (a) denotes control input vectors, a (t) denotes fuzzy sets, and q ═ 1, 2., κ denotes control input vectorsi,BiAnd CiAre all a matrix of the system, and,is a state rate of change vector, y (t) is a system output vector, for the purposes of an n-dimensional euclidean space, is an m-dimensional euclidean space;
the global model corresponding to the controlled object is as follows:
in the formula, wi(x (t)) is degree of membershipCorresponding fuzzy weight in the controlled object;
the description mode of the observer is as follows:
for observer rule j, when observer blurs precondition variableCorresponding fuzzy set dependencesIs next toThen, obtaining:
in the formula (I), the compound is shown in the specification,in order to observe the rate of change of state vector,is a state vector of an observer, anu (t) is a control input vector, observer rule j satisfies j is more than or equal to 2 and less than or equal to r,is a fuzzy precondition of the observerA corresponding fuzzy set, q 1,2jAs observer parameters, Aj,BjAnd CjAre all a matrix of the system, and,an input signal of an observer being an adaptive event triggering mechanism;
the global model corresponding to the observer is as follows:
in the formula (I), the compound is shown in the specification,is degree of membershipA corresponding fuzzy weight in the observer;
the controller is described in the following manner:
for controller rule l, when the controller fuzzes the precondition variablesThe corresponding fuzzy sets are sequentiallyThen, obtaining:
wherein u (t) is a control input vector,fuzzy precondition variables for a controllerA corresponding fuzzy set, q 1, 2., p, l 1, 2., r, p is the number of fuzzy preconditions of the controller, KlIn order to be a matrix of fuzzy controller parameters,inputting a state vector for a controller;
the global model corresponding to the controller is as follows:
in the formula (I), the compound is shown in the specification,is the fuzzy weight of the fuzzy controller;
based on the controlled object, the observer and the global model corresponding to the controller, the closed-loop model of the nonlinear network control system is obtained as follows:
in the formula (I), the compound is shown in the specification,to observe the coupling vectors for the rate of change of state and the state error,
is the difference between the current observer state and the last observer state triggered, ey(T) is the error of the current output signal of the T-S fuzzy model and the last transmitted output signal;
therein, Ψijl、Aijl、Lijl、LeijlAnd LeyjIntermediate parameters in a closed-loop model of the nonlinear network control system are controlled.
2. The double-ended event triggered nonlinear control method according to claim 1, characterized in that in a nonlinear networked control system closed-loop model, a first event generator is provided in a sensor-to-observer data transmission channel under an adaptive event triggering mechanism;
in the sensor-to-observer data path, at event trigger timesSystem output signal transmitted by first event generator at sensor endAt the time of dayThe data center of the observer is reached through a transmission network; wherein the content of the first and second substances,is a time of dayThe network at the sensor end of (1) induces a time delay,τyis a time of dayThe maximum network induced delay of the sensor end;
the adaptive event trigger mechanism determines the current signal based on adaptive event trigger conditionsWhether or not to be transmitted to the observer, the next event triggering instant when the adaptive event triggering condition is metComprises the following steps:
in the formula (I), the compound is shown in the specification,to satisfy the conditionThe set of (a) and (b),in order for the number of data lost packets to occur continuously,in order for the maximum allowable amount of data packet loss to continuously occur,for the transposition of the difference between the current output signal and the last trigger instant, ΩyIs an event trigger matrix, and Ωy>0,ey(T) is the error, χ, between the current output signal of the T-S fuzzy model and the last transmitted output signaly(t) is satisfying xy(t)∈(0,1]Variable trigger threshold of yT() means for outputting a vector for the current system, y (-) is the current system output vector;
wherein the error e of the current output signal of the T-S fuzzy model and the last transmitted output signaly(t) is:
variable trigger threshold χy(t) satisfies:
wherein the content of the first and second substances,in order for the event to trigger the threshold value,for adjusting the variable trigger threshold χy(t) a parameter of convergence speed;
wherein d (t) is a time-varying function, in order to be able to process the time interval,epsilon is an intermediate parameter which is,tau is the maximum network-induced time delay,τyis a time of dayThe maximum network at the sensor end of (1) induces a time delay,is a time of dayThe maximum network at the observer end of (1) induces a time delay.
3. The double-ended event triggered nonlinear control method according to claim 2, characterized in that a second event generator is provided in the observer to controller data transmission channel;
in the data transmission channel from the observer to the controller, at the moment of application of the triggerObserver end
Observer state signal transmitted by second event generatorAt the time of dayThe data center of the controller is reached through a transmission network;is a time of dayThe maximum network at the observer end of (1) induces a time delay, is a time of dayThe maximum network induced delay of the observer end;
the second event generator reduces the network redundancy of the state signal under the condition of an adaptive event triggering mechanism according to the state information of the observer, wherein the condition of the adaptive triggering mechanism is as follows:
in the formula (I), the compound is shown in the specification,for the moment when the next event trigger occurs,as a transpose of the difference of the current observer state and the last triggered observer state,in order to trigger the matrix for an event,the difference between the current observer state and the last triggered observer state,to adapt the trigger threshold of the trigger mechanism,is a transpose of the state vector of the current observer,is the current observer state vector;
in the formula (I), the compound is shown in the specification,in order for the event to trigger the threshold value,for adjusting trigger thresholdA parameter of convergence speed;
4. the double-ended event triggered nonlinear control method according to claim 3, wherein the step S2 is specifically:
s21, defining a Lyapunov function containing event triggering information;
s22, carrying out derivation on the Lyapunov function, and eliminating an integral term in the Lyapunov function to obtain the stability condition of the closed-loop model of the nonlinear networked control system;
s23, eliminating a nonlinear term in the stability condition of the nonlinear networked control system closed-loop model;
and S24, on the basis of the stability condition of the networked control system closed-loop model with the nonlinear term eliminated, obtaining a system stability condition related to the membership function based on the linear matrix inequality by using the time delay information of the membership function, and taking the system stability condition as the final stability condition of the nonlinear networked control system closed-loop model.
5. The double-ended event triggered nonlinear control method according to claim 4, wherein the Lyapunov function in step S21 is:
V(t)=V1(t)+V2(t)+V3(t)
wherein ξT(t) is the transpose of the coupling vector for observed state change rate and state error, ξ (t) is the coupling vector for observed state change rate and state error, P, Q, R and W are both matrices with appropriate dimensions, ikh is an intermediate parameter which is a function of,dMis the upper delay bound.
6. The double-ended event triggered nonlinear control method according to claim 5, wherein in step S22, when d is satisfiedM>0,Andif there is a matrix P with suitable dimensions>0,Q>0,R>0,W>0,Ωy>0,And U, the stability condition of the closed-loop model of the nonlinear networked control system is as follows:
in the formula phiijlIn order to determine the matrix for the stability,andare all phiijlIs an element ofA transposed matrix of mid-diagonal positions;
wherein
F=[Aijl Lijl 0 Leijl Leyj]
In the formula (I), the compound is shown in the specification,andare all made ofThe element in (1), col {. is a column matrix, diag {. is a block diagonal matrix, and F is an intermediate matrix;
and is
In the formula, CiAre all system matrices, G ═ I I],E=[I 0]And I is an identity matrix of a suitable dimension.
7. The double-ended event triggered nonlinear control method according to claim 6, characterized in that the stepsIn step S24, when d isM>0,Andand the membership function of the fuzzy controller satisfiesWherein 0<ιl≤1,ιlFor appropriate constants, if any, with matrix X of appropriate dimensions>0, TjAnd YlThen, the system stability condition related to the membership function based on the linear matrix inequality is:
wherein the content of the first and second substances,iota discrimination matrix for stability after elimination of non-linear termsjIs less than iotalIs suitably constant;
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